When Do You Flip the Inequality Sign?

You're sailing through your homework then...huh. An inequality with lots of negatives and absolute values. Help! When do you flip the inequality sign?

No fear! There are a couple of occasions when you flip the inequality, and we'll go through them below.

TL;DR (Too Long; Didn't Read)

TL;DR (Too Long; Didn't Read)

Flip the inequality sign when you multiply or divide both sides of an inequality by a negative number.

You also often need to flip the inequality sign when solving inequalities with absolute values.

Multiplying and Dividing Inequalities by Negative Numbers

The main situation where you'll need to flip the inequality sign is when you multiply or divide both sides of an inequality by a negative number.

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For example, consider the following problem:

3_x_ + 6 > 6_x_ + 12

To solve, you need to get all the x-es on the same side of the inequality. Subtract 6_x_ from both sides in order to only have x on the left.

3_x_ −6_x_ + 6 > 6_x_ −6_x_ + 12

−3_x_ + 6 > 12

Now isolate the x on the left side by moving the constant, 6, to the other side of the inequality. To do this, subtract 6 from both sides.

− 3_x_ + 6 − 6 > 12 − 6

−3_x_ > 6

Now divide both sides of the inequality by −3. Since you're dividing by a negative number, you need to flip the inequality sign.

−3_x_ (÷ −3) < 6 (÷ − 3)

x < − 2.

The same rule would apply if you're multiplying both sides by a fraction. Multiplying and dividing are inverses of the same process, kind of like adding and subtracting, so the same rules apply to both.

Absolute Value Problems

You also need to think about flipping the inequality sign when you're dealing with absolute value problems.

Take the following example. If you have:

| 3_x_ | + 6 < 12,

Then first of all you want to isolate the absolute value expression on the left side of the inequality (it makes life easier). Subtract 6 from both sides to get:

| 3_x_ | < 6.

Now, you need to rewrite this expression as a compound inequality. | 3_x_ | < 6 can be written in two ways:

3_x_ < 6 (the "positive" version), or

3_x_ > −6 (the "negative" version).

These two statements can also be written in a single line:

−6 < 3_x_ < 6.

The output of an absolute value expression is always positive, but the "x" inside the absolute value signs might be negative, so we need to consider the case when x is negative. We're essentially multiplying by −1: we're multiplying x by negative one on the left (but since it's inside absolute value signs the outcome is still positive), and then we're multiplying the right side by negative one and switching the inequality sign because we just multiplied by a negative.

That gives us our two inequalities (or our "compound inequality"). We can easily solve both of them.

3_x_ < 6 becomes x < 2 once we divide both sides by 3.

3_x_ > −6 becomes x > −2 after we divide both sides by 3.

So the solution is x < 2 and x > −2, or −2 < x < 2.

These kinds of problems take some practice, so don't worry if you aren't getting it at first! Keep at it and it will eventually become second nature.